Apart from the accurate description of the wavefunction entanglement encoded in the pump-probe spectrum, the self-consistent Eq. (REF ) also provides an alternative way to interpret the light-induced spin fluctuation dynamics. We note that the leading order \(m=1\)  term in the series of Eq. (REF ) is the second-order time derivative of \(f_\mathrm {Q}\) , which reflects an inertia of the underlying wavefunction toward the change. This inertia enables self-driving of spin fluctuations and explains why the QFI in Fig. REF  continues to oscillate for \(t \gtrsim 5 t_\mathrm {h}^{-1}\)  even when the pump field vanishes. In contrast, trRIXS spectra for non-interacting fermions, and accordingly any quantities extracted from it, completely recovers to its equilibrium counterpart after the pump ends [1]}, [2]}. Therefore, the self-driving wavefunctions are unique for correlated systems with interactions. Including these high-order time-derivatives in Eq. (REF ) is crucial to correctly depict the QFI of nonequilibrium wavefunctions. As a result, the “snapshot QFI” \(\bar{f}_\mathrm {Q}(q, t)\)  serves as a good approximation for the \(f_\mathrm {Q}\)  at the pump arrival, but starts to deviate from each other at later time delays where the pump tails off.
